Efficient mesh freee methods for plate analysis

Effective Mesh-free Methods for Plate Analysis Wu Wenxin NATIONAL UNIVERSITY OF SINGAPORE 2007 Effective Mesh-free Methods for Plate Analysis Wu Wenxin (B. Eng., Shanghai Jiaotong University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 To Shu Rong and Jia Xi Acknowledgements I would like to express my deepest gratitude to my supervisors, Professor C. Shu and Professor C. M. Wang, for their invaluable guidance, encouragement and patience throughout this study. My gratitude also extends to my wife and daughter for their support and encouragement over my PhD candidature period. Finally, I wish to thank the National University of Singapore for providing me with a research scholarship, which makes this study possible. Wu Wenxin I Table of Contents Acknowledgements I Table of Contents II Summary XI List of Tables .XIII List of Figures XVII Notations XXV Abbreviations XXVIII Chapter Introduction . 1.1 Background of Plate Analysis . 1.1.1 Introduction to Plate Theories 1.1.2 Analytical Methods for Plate Analysis 1.1.3 Numerical Methods for Plate Analysis 1.2 Literature Review on Mesh-free Methods 1.2.1 Disadvantages of Traditional Numerical Methods 1.2.2 Concept of Mesh-free 11 1.2.3 Classification of Mesh-free Methods . 13 1.2.3.1 A Particle Method: Smoothed Particle Hydrodynamics (SPH) 17 1.2.3.2 Mesh-free Methods of Integral Type 18 1.2.3.3 Mesh-free Methods of Non-Integral Type 20 1.2.4 Desirable Mesh-free Methods for Plate Analysis 24 1.3 Objectives of Thesis 25 II 1.4 Organization of Thesis 25 Chapter LSFD Method and LRBFDQ Method 29 2.1 Least Squares-based Finite Difference (LSFD) Method 29 2.1.1 Conventional Finite Difference Method (FDM) 29 2.1.2 Least Squares-based Finite Difference (LSFD) Method . 30 2.1.2.1 Formulas for Derivative Discretization 30 2.1.2.2 Chain Rule for Discretization of Derivatives 37 2.1.2.3 LSFD Formulas in Local (n, t)-Coordinates at Boundary 39 2.1.2.4 Numerical Analysis of a Sample PDE Using LSFD . 41 2.1.3 Function Value Problems and Eigenvalue Problems . 45 2.1.4 Concluding Remarks 46 2.2 Local Radial Basis Function-based Differential Quadrature (LRBFDQ) Method . 47 2.2.1 Radial Basis Functions (RBFs) and Interpolation Using RBFs . 47 2.2.2 Traditional RBF-based Schemes for Solving PDEs 50 2.2.3 Local Radial Basis Function-based Differential Quadrature (LRBFDQ) Method . 51 2.2.3.1 Formulas for Derivative Discretization 51 2.2.3.2 LRBFDQ Formulas in Local (n, t)-Coordinates at Boundary 54 2.2.3.3 Numerical Analysis of Sample PDEs Using LRBFDQ 55 2.2.4 Concluding Remarks 72 III Chapter Applications of LSFD and LRBFDQ for Solving Helmholtz Equations . 73 3.1 TM Modes and TE Modes in Metallic Waveguides – Application of LSFD . 74 3.1.1 Definition of Problem 74 3.1.2 Numerical Algorithm . 75 3.1.2.1 Numerical Discretization by LSFD 75 3.1.2.2 Dealing with Singular Points on Boundary Γ 78 3.1.3 Results and Discussion 79 3.1.3.1 Rectangular Waveguide 79 3.1.3.2 Double-Ridged Waveguide . 81 3.1.3.3 L-Shaped Waveguide 82 3.1.3.4 Single-Ridged Waveguide 83 3.1.3.5 Coaxial Rectangular Waveguide . 85 3.1.3.6 Vaned Rectangular Waveguide . 86 3.1.4 Concluding Remarks 87 3.2 Free Vibration of Uniform Membranes – Application of LRBFDQ 87 3.2.1 Definition of Problem 87 3.2.2 Numerical Discretization by LRBFDQ . 88 3.2.3 Results and Discussion 89 3.2.3.1 Circular Membrane . 90 3.2.3.2 Rectangular Membrane . 92 3.2.3.3 Half Circle+Triangle Membrane . 94 3.2.3.4 L-Shaped Membrane . 96 3.2.3.5 Concave Membrane with High Concavity 98 IV 3.2.3.6 Multi-Connected Membrane . 101 3.2.4 Concluding Remarks 103 Chapter Plate Theories and Numerical Implementation . 105 4.1 Thin Plate Theory for Small Deflection Problems . 105 4.1.1 Displacement Components 105 4.1.2 Strain-Displacement Relations . 106 4.1.3 Stresses and Stress Resultants 107 4.1.4 Differential Equation for Transversely Loaded Plates . 109 4.1.5 Differential Equation for Freely Vibrating Plates 113 4.1.6 Differential Equation for Buckling of Plates . 114 4.1.6.1 Plates under Combined Transverse and In-Plane Loads . 114 4.1.6.2 Buckling of Plates . 117 4.1.7 Boundary Conditions . 118 4.1.8 Numerical Implementation 120 4.1.8.1 Discretization of Governing Equations . 120 4.1.8.2 Implementation of Boundary Conditions 123 4.2 Thin Plate Theory for Large Deflection Problems . 133 4.2.1 Bending Equations of Plates with Large Deflections 133 4.2.2 Equations of Motion for Large-Amplitude Free Vibration of Thin Plates 136 4.2.3 Boundary Conditions . 137 4.2.4 Numerical Implementation 137 4.2.4.1 Bending of Plates with Large Deflections 137 V 4.2.4.2 Large Amplitude Free Vibration of Plates 138 4.3 Shear Deformable Plate Theory for Small Deflection Problems 143 4.3.1 Displacement Components 144 4.3.2 Strain-Displacement Relations . 144 4.3.3 Stress Resultant-Displacement Relations 145 4.3.4 Governing Equations of Motion 146 4.3.5 Governing Equations for Bending . 147 4.3.6 Governing Equations for Free Vibration . 148 4.3.7 Boundary Conditions . 149 4.3.8 Numerical Implementation Using LSFD Method 151 4.3.8.1 Discretization of Governing Equations . 151 4.3.8.2 Implementation of Boundary Conditions 152 Chapter LSFD for Thin Plate Vibration 158 5.1 Small-Amplitude Free Vibration of Thin Plates with Arbitrary Shapes . 159 5.1.1 Simply Supported and Clamped Plates . 160 5.1.1.1 Introduction 160 5.1.1.2 Results and Discussion . 162 • Symmetric Trapezoidal Plates 162 • Symmetric Parabolic Trapezoidal Plates 162 • Rhombic Plates . 164 • Sectorial Plates 165 • Circular and Elliptical Plates 167 VI • Annular Plates . 170 5.1.1.3 Concluding Remarks . 170 5.1.2 Completely Free Plates 172 5.1.2.1 Introduction . 172 5.1.2.2 Problem Definition and Numerical Solution 173 5.1.2.3 Results and discussion 174 • Frequencies of Circular and Elliptical Plates 175 • Frequencies of Lifting-Tab Shaped and 45o Right Triangular Plates 176 • Verification of Radii of Nodal Circles of the Circular Plate 178 • Verification of Natural Boundary Conditions . 179 • Distributions of Mode Shapes and Modal Stress Resultants 183 • Peak Values of Modal Deflections and Modal Stress Resultants . 191 5.1.2.4 Concluding Remarks . 193 5.2 Large-Amplitude Free Vibration of Thin Plates with Arbitrary Shapes . 194 5.2.1 Motivation and Literature Review . 194 5.2.2 Results and discussion . 196 5.2.2.1 Square Plates . 197 5.2.2.2 Circular Plates . 201 5.2.2.3 L-Shaped Plate 202 5.2.2.4 Square Plates with Semi-Circular Edge Cuts 204 5.2.3 Concluding Remarks 206 VII References 285 Melenk, J. M. and Babuska, I. (1996). “The partition of unity finite element method: Basic theory and applications,” Comput. Methods Appl. Mech. Engrg., 139, 289-314. Micchelli, C. A. (1986). “Interpolation of scattered data: distance matrices and conditionally positive definite functions,” Constr. Approx., 2, 11-22. Mindlin, R. D. (1951). “Influence of rotatory inertia and shear on flexural motion of isotropic elastic plates,” Journal of Applied Mechanics, ASME, 18, 31-38. Mizusawa, T. and Takami, T. (1992). “Vibration of tapered thickness annular sector plates by spline strip method,” J. Sound Vibr., 154, 147-160. Mizusawa, T., Kajita, T., Naruoka, M. (1980a). “Buckling of skew plate structures using B-spline functions,” International Journal for Numerical Methods in Engineering, 15, 87-96. Mizusawa, T., Kajita, T., Naruoka, M. (1980b). “Analysis of skew plate problems with various constraints,” Journal of Sound and Vibration, 73, 575-584. Mizusawa, T., Kajita, T. (1986). “Vibration and buckling of skew plates with edges elastically restrained against rotation,” Computers and Structures, 22, 987-994. Mizusawa, T. and Leonard, J. W. (1990). “Vibration and buckling of plates with mixed boundary conditions,” Eng. Struct., 12, 285-290. Monaghan, J. J. (1982). “Why particle methods work,” SIAM J. Sci. Stat. Comput., 3(4), 422. Monaghan, J. J. (1988). “An introduction to SPH,” Comput. Phys. Comm., 48, 89-96. Montgomery, J. P. (1971). “On the complete eigenvalue solution of ridged waveguide,” IEEE Trans. Microwave Theory Tech., MTT-19, 547-555. References 286 Narita, Y. (1985). “Natural frequencies of free, orthotropic elliptical plates,” J. Sound Vib., 100, 83-89. Nayroles, B., Touzot, G. and Villon, P. (1992). “Generalizing the finite element method: Diffuse approximation and diffuse elements,” Computational Mechanics, 10, 307318. Oden, J. T. (1972). Finite Elements of Nonlinear Continua, McGraw-Hill, New York. Onate, E., Idelsohn, S., Zienkiewicz, O. C. and Taylor, R. L. (1996a). “A finite point method in computational mechanics. Applications to convective transport and fluid flow,” Int. J. Numer. Meth. Engng., 39, 3839-3866. Onate, E., Idelsohn, S., Zienkiewicz, O. C., Taylor, R. L. and Sacco, C. (1996b). “A stabilized finite point method for analysis of fluid mechanics problems,” Comput. Methods Appl. Mech. Engrg., 139, 315-346. Prabhu, M. S. S. and Durvasula, S. (1972). “Stability of clamped skew plates,” Appl. Sci. Res., 26, 255-271. Rao, G. V., Raju, I. S. and Raju, K. K. (1976a). “A finite element formulation for large amplitude flexural vibrations of thin rectangular plates,” Computers and Structures, 6, 163-167. Rao, G. V., Raju, K. K. and Raju, I. S. (1976b). “Finite element formulation for the large amplitude free vibrations of beams and orthotropic circular plates,” Computers and Structures, 6, 169-172. Rashed, Y. F., Aliabadi, M. H. and Brebbia, C. A. (1999). “Transformation of domain integrals in BEM for thick foundation plates,” Journal of Engineering Mechanics, 125(9), 1062-1070. References 287 Rayleigh, J. W. (1877). Theory of Sound, Macmillan Vol. (reprinted by Dover Publications, 1945). Reddy, J. N. (1993). An introduction to the finite element method, 2nd Ed., McGraw-Hill, New York. Reddy, J. N. (1999). Theory and Analysis of Elastic Plates, 1st Ed., Taylor & Francis. Ritz, W. (1909). “Uber eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik,” Journal für Reine und Angewandte Mathematik, 135, 1-61. Salvadori, M. G. (1949). “Numerical computation of buckling loads by finite differences,” Proceedings of ASCE, 75, 1441-1475. Sarkar, T. K., Athar, K., Arvas, E., Manela, M. and Lade, R. (1989). “Computation of the propagation characteristics of TE and TM modes in arbitrarily shaped hollow waveguides utilizing the conjugate gradient method,” J. Electromagn. Waves Appl., 3, 143-165. Sato, K. (1973). “Free flexural vibration of an elliptical plate with free edge,” Journal of the Acoustical Society of America, 54, 547-550. Shi, Y. and Mei, C. (1996). “A finite element time domain modal formulation for large amplitude free vibrations of beams and plates,” Journal of Sound and Vibration, 193, 453-464. Shu, C. (2000). Differential quadrature and its application in engineering, SpringerVerlag London Limited. Shu, C. and Chew, Y. T. (1999). “Application of multi-domain GDQ method to analysis of waveguides with rectangular boundaries,” Progress In Electromagnetics Research, PIER 21, 1-19. References 288 Shu, C., Ding, H. and Yeo, K. S. (2003). “Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations,” Comput. Methods Appl. Mech. Engrg., 192, 941-954. Shu, C., Wu, W. X., Wang, C. M. (2005). “Analysis of metallic waveguides by using least squares-based finite difference method,” Computers, Materials, & Continua, 2(3), 189-200. Singh, B. and Chakraverty, S. (1992). “On the use of orthogonal polynomials in the Rayleigh-Ritz method for the study of transverse vibration of elliptic plates,” Comput. Struct., 43, 439-443. Swaminathan, M., Arvas, E., Sarkar, T. K. and Djordjevic, A. R. (1990). “Computation of cutoff wavenumbers of TE and TM modes in waveguides of arbitrary cross sections using a surface integral formulation,” IEEE Trans. Microwave Theory Tech., MTT-38, 154-159. Szilard, R. (1972). Theory and Analysis of Plates, Chap. 3, Prentice Hall, Upper Saddle River, N.J. Tezcan, S. S. (1967). “Nonlinear Analysis of Thin Plates by Framework Method,” AIAA Journal, 5(10), 1890-1892. Timoshenko, S. P. and Woinowsky-Krieger, S. (1959). Theory of Plates and Shells, 2nd Edition, McGraw-Hill, New York. Ugural, A. C. (1999). Stresses in Plates and Shells, 2nd Edition, McGraw-Hill. Utsumi, Y. (1985). “Variational analysis of ridged waveguide mode,” IEEE Trans. Microwave Theory Tech., MTT-33, 111-120. References 289 Utsunomiya, T., Watanabe, E., Eatock Taylor, R. (1998). “Wave response analysis of a box-like VLFS close to a breakwater,” 17th International Conference on Offshore Mechanics and Arctic Engineering, ASME, OMAE98-4331, 1-8. Ventsel, E. S. (1997). “A posteriori error estimate analysis for BEM in plate-bending problems,” Mathematics and Mechanics of Solids, 2(3), 321-333. Voth, T. E. and Christon, M. A. (2001). “Discretization errors associated with reproducing kernel methods: one-dimensional domains,” Comput. Methods Appl. Mech. Engrg., 190, 2429-2446. Wagner, G. J. and Liu, W. K. (2000). “Turbulence simulation and multiple scale subgrid models,” Computational Mechanics, 25, 117-136. Wah, T. (1963). “Large amplitude flexural vibration of rectangular plates,” Int. J. Mech. Sci., 5, 425-438. Wang, C. M., Kitipornchai, S., Xiang, Y., Liew, K. M. (1993). “Stability of skew Mindlin plates under isotropic in-plane pressure,” Journal of Engineering Mechanics, ASCE, 119, 393-401. Wang, C. M., Liew, K. M., Alwis, W. A. M. (1992). “Buckling of skew plates and corner condition for simply supported edges,” Journal of Engineering Mechanics, ASCE, 118, 651-662. Wang, C. M., Reddy, J. N. and Lee, K. H. (2000a). Shear Deformable Beams and Plates, Relationships with Classical Solutions, 1st Edition, Elsevier, Oxford. Wang, C. M., Utsunomiya, T. and Watanabe, E. (2000b). “On obtaining accurate stress resultants in vibration analysis of floating plate structures,” Journal of The Institution of Engineers, 40(5), 31-42. References 290 Wang, C. M., Xiang, Y., Utsunomiya, T., Watanabe, E. (2001). “Evaluation of modal stress resultants in freely vibrating plates,” International Journal of Solids and Structures, 38(36-37), 6525-6558. Wang, C. M., Xiang, Y., Watanabe, E., Utsunomiya, T. (2004). “Mode shapes and stressresultants of circular Mindlin plates with free edges,” Journal of Sound and Vibration, 276(3-5), 511-525. Wang, G. and Cheng-Tzu, T.H. (1994). “Static and dynamic analysis of arbitrary quadrilateral flexural plates by B3-spline functions,” Int. J. Solids Struct., 31, 657667. Wang, X., Tan, M., Zhou, Y. (2003). “Buckling analysis of anisotropic plates and isotropic skew plates by the new version differential quadrature method,” ThinWalled Structures, 41, 15-29. Wang, X. X., Qian, J. and Huang, M. K. (1991). “A boundary integral equation formulation for large amplitude nonlinear vibration of thin elastic plates,” Computer Methods in Applied Mechanics and Engineering, 86, 73-86. Watanabe, E., Wang, C. M., Utsunomiya, T. and Moan, T. (2004). “Very Large Floating Structures: Applications, Analysis and Design,” CORE Report No. 2004-02, Centre for Offshore Research and Engineering, National University of Singapore. Williams, M. L. (1952a). “Surface stress singularities resulting from various boundary conditions in angular corners of plates under bending,” Proceedings of the First US National Congress of Applied Mechanics. References 291 Williams, M. L. (1952b). “Stress singularities resulting from various boundary conditions in angular corners of plates in externsion,” Journal of Applied Mechanics, 19, 526528. Wittrick, W. H. (1953). “Buckling of oblique plates with clamped edges under uniform compression,” Aeronautical Quarterly, 4, 151-163. Wittrick, W. H., Williams, F. W. (1974). “Buckling and vibration of anisotropic or isotropic plate assemblies under combined loadings,” International Journal of Mechanical Sciences, 16, 209-239. Wu, Z. M. (1992). “Hermite-Birkhoff interpolation of scattered data by radial basis functions,” Approximation Theory and its Application, 8, 1-10. Xu, S. L. (1995). Fortran programs for common algorithms, 2nd edition (in Chinese), Qinghua University Press. Yamaki, N. (1961). “Influence of large amplitudes on flexural vibrations of elastic plates,” ZAMM, 41, 501-510. York, C. B., Williams, F. W. (1994). “Theory and buckling results for infinitely wide, stiffened skew plate assemblies,” Composite Structures, 28, 189-200. York, C. B., Williams, F. W. (1995). “Buckling analysis of skew plate assemblies: classical plate theory results incorporating Lagrangian multipliers,” Computers and Structures, 56, 625-635. Young, W. C. (1989). Roark’s Formulas for Stress and Strain, 6th ed., Sec. 10.11, McGraw-Hill, New York. References 292 Zerroukat, M., Power, H. and Chen, C. S. (1998). “A numerical method for heat transfer problems using collocation and radial basis functions,” Int. J. Numer. Meth. Engng., 42, 1263-1278. Zhang, L. T., Wagner, G. J. and Liu, W. K. (2002). “A parallelized mesh-free method with boundary enrichment for large-scale CFD,” Journal of Computational Physics, 176, 483-506. Zhang, X., Liu, X. H., Song, K. Z. and Lu, M. W. (2001). “Least-squares collocation meshless method,” Int. J. Numer. Meth. Engng., 51, 1089-1100. Zhu, T. L., Zhang, J. D. and Atluri, S. N. (1999). “A meshless numerical method based on the local boundary interal equation (LBIE) to solve linear and non-linear boundary value problems,” Engineering Analysis with Boundary Elements, 23, 375389. Zienkiewitcz, O. C. and Taylor, R. I. (1991). The Finite Element Method, 4th ed., Vol. (Solid and Fluid Mechanics, Dynamics and Non-linearity), McGraw-Hill, London. Appendix An Alternative Expression for Natural Boundary Condition of Simply Supported Edge of Thin Plates This alternative expression for the natural boundary condition of a simply supported edge of thin plates replaces the zero normal bending moment condition at the edge. It features a reduction of the Laplacian operation of the plate deflection to a first-order derivative of the deflection, a form useful for implementation in the LSFD method or the LRBFDQ method for plate analysis. Introduction According to the classical thin plate theory (Timoshenko and Woinowsky-Krieger 1959), the boundary conditions for a simply supported edge are w=0 (1a) ∂2w ∂2w + ν =0 ∂n ∂t (1b) in which w is the transverse deflection, ν the Poisson ratio, n and t the normal and tangential coordinates to the plate edge. Equation (1a) implies that the deflection w ( x, y ) at the plate edge is zero, and Eq. (1b) implies that the bending moment ( ) M n = − D ∂ w ∂n + ν ⋅ ∂ w ∂t normal to the edge is zero. 293 Appendix 294 For simply supported edges that are straight, it is well known that the natural boundary condition given by Eq. (1b) may be expressed alternatively (Timoshenko and Woinowsky-Krieger 1959; Conway 1960) as ∇2w = (2) where ∇ is the Laplacian operator. Eq. (2) may be readily derived from the fact that since w = along the plate edge, then ∂ w ∂t = holds for a rectilinear edge and hence ∂ w / ∂n = from Eq. (1b). Physically, Eq. (2) may be interpreted as that the sum of moments (i.e. M n + M t ) is zero for a straight simply supported edge. This alternative form allows direct implementation of the boundary conditions in the finite difference method when compared to the form given in Eq. (1b) since the former may be substituted into the discretized governing equations directly (see Eq. (4.45-1)). Alternative Expression for Eq. (1b) for Curved Simply Supported Edge Eq. (2) has to be modified for a simply supported edge that is not straight but curved. The general alternative expression for the natural boundary condition involves the moment sum and a lower derivative for easy implementation in the LSFD or LRBFDQ method. This alternative form is given by ∇2 w = ± −ν ∂w ⋅ r ∂n (3) where r is the radius of curvature of the plate boundary curve at a boundary point concerned, which is always taken as positive value. The ‘ + ’ sign should be used for the convex boundary (e.g. the curve ABC in Fig. A1), and the ‘ − ’ sign should be used for the concave boundary (e.g. AB1C in Fig. A1). Note that Eq. (3) reduces the second-order Appendix 295 derivative operator ∇ w at the boundary to a first-order derivative, which is easy for numerical implementation. And for a straight edge ∇ 2W = since r = ∞ . Proof for Eq. (3) The proof of Eq. (3) is given as follows. Using Eq. (1b), we have ∂2w ∂2w  ∂2w ∂2w  ∂2w ∂2w ∇ w = + =  +ν  + (1 −ν ) = (1 −ν ) ∂n ∂t ∂t  ∂t ∂t  ∂n (4) Along the plate edge, which is a curve in the x-y plane and can generally be defined by an equation F ( x, y ) = , the slope of the tangent of the curve in the x-y plane is yx = − Fx Fy (5) and the second order derivative of y to x is yxx = d  Fx − dx  Fy ∂ F   = −   x   ∂x  Fy  ∂ F  +  x  ∂y  Fy    y x    (6) F ⋅F − F ⋅F  Fxy ⋅ Fy − Fx ⋅ Fyy xx y x xy  =− + ⋅ yx  2   ( Fy ) ( Fy )     F F  Fxx − x ⋅ Fxy + Fxy ⋅ y x − x ⋅ Fyy ⋅ y x  Fy Fy   =− Fy =−  Fxx + Fxy ⋅ yx + Fyy ⋅ ( y x )    Fy From the simply supported boundary condition, we know that for all the points ( x, y ) ∈ {( x, y ) | F ( x, y ) = 0} , the equation w ( x, y ) = is satisfied. Therefore, a general relation between the functions F ( x, y ) and w ( x, y ) can be assumed as Appendix 296 w ( x, y ) = g ( x, y )  F ( x, y )  ξ (7) where g ( x, y ) is a general continuous differentiable function and ξ > . From Eq. (7), we have wx = g x F ξ + ξ gF ξ −1 Fx (8a) wy = g y F ξ + ξ gF ξ −1 Fy (8b) wxx = g xx F ξ + 2ξ g x F ξ −1 Fx + ξ (ξ − 1) gF ξ − ( Fx ) + ξ gF ξ −1 Fxx (8c) wxy = g xy F ξ + ξ g x F ξ −1 Fy + ξ g y F ξ −1 Fx + ξ (ξ − 1) gF ξ − Fx Fy + ξ gF ξ −1 Fxy (8d) wyy = g yy F ξ + 2ξ g y F ξ −1 Fy + ξ (ξ − 1) gF ξ − ( Fy ) + ξ gF ξ −1 Fyy (8e) Along the boundary curve, from Eqs. (5), (8a-e), and the condition F ( x, y ) = , we have wx Fx = wy Fy (9a)   2 wxx + wxy ⋅ yx + wyy ⋅ ( yx )  = Fxx + Fxy ⋅ y x + Fyy ⋅ ( y x )      wy Fy (9b) So, Equations (5), (6) and (9a,b) give yx = − wx wy (10)  wxx + wxy ⋅ y x + wyy ⋅ ( y x )    wy (11) yxx = − As shown in Fig. A1, x-y is a global coordinate system whereas n-t is the local coordinate system. The n- and t - axes are normal and tangential to the boundary curve of the plate. We denote by θ the angle between positive x- axis and positive n- axis. Then Appendix 297 the slopes of the t - axis and the n- axis in the x-y coordinate system are y x and tan θ , respectively. Since they are perpendicular to each other, we have y x ⋅ tan θ = −1 i.e. yx = − cot θ (12) On the other hand, for the plate deflection w ( x, y ) , we have the relation ∂w = wx cos θ + wy sin θ ∂n (13) In view of Eqs. (10) to (13), we have yxx ⋅ 32 1 + ( yx )    =− =− =− =− ∂w ∂n (14)  ⋅ ( wx cos θ + wy sin θ ) wxx + wxy ⋅ y x + wyy ⋅ ( yx )  ⋅   32 wy   1+ y  ( x)  wxx + wxy ( − cot θ ) + wyy cot θ ( + cot θ ) 32 ⋅ ( − y x cos θ + sin θ ) cos θ cos θ + wyy ⋅ sin θ sin θ ⋅ ( cot θ cos θ + sin θ ) 32  cos θ  1 +   sin θ  wxx − wxy ⋅ wxx sin θ − wxy sin θ cos θ + wyy cos θ ⋅ sin θ ⋅ sin θ sin θ sin θ ∂ w sin θ =− ⋅ ∂t sin θ From Eqs. (4) and (14), we have Appendix 298 ∇ w = − (1 −ν ) yxx 32 1 + ( yx )    ⋅ ∂w sin θ ⋅ ∂n sin θ (15) Eq. (15) can be further simplified by considering the following situations: 1) If a section of the boundary curve is at the underside of the domain and is curved outwards (eg ABC in Fig. A1), we have sin θ ≤ and ∇2w = yxx 32 1 + ( y x )    = , then r −ν ∂w ⋅ . r ∂n 2) If a section of the boundary curve is at the underside of the domain and is curved inwards (eg AB1C in Fig. A1), we have sin θ ≤ and ∇2 w = − yxx = − , then r 32 1 + ( yx )    −ν ∂w ⋅ . r ∂n 3) If a section of the boundary curve is at the upside of the domain and is curved outwards (eg DEF in Fig. A1), we have sin θ ≥ and ∇2w = yxx 32 1 + ( yx )    =− , then r −ν ∂w ⋅ . r ∂n 4) If a section of the boundary curve is at the upside of the domain and is curved inwards (eg DE1 F in Fig. A1), we have sin θ ≥ and ∇2 w = − yxx 32 1 + ( y x )    −ν ∂w ⋅ . r ∂n In summary, we have proven expression (3) and the discussions that followed. = , then r Appendix 299 y t E F n D E1 B1 A C B o x Fig. A1 x-y coordinate system and n-t coordinate system. List of Publications 1. Shu, C., Wu, W. X. and Wang, C. M. (2005). “Analysis of metallic waveguides by using least square-based finite difference method,” CMC: Computers, Materials and Continua, 2(3), 189-200. 2. Wang, C. M., Wu, W. X., Shu, C. and Utsunomiya, T. (2005). “LSFD method for vibration analysis of plates with free edges,” Advances in Steel Structures, ICASS’05, 1715-1722. 3. Wu, W. X., Shu, C. and Wang, C. M. (2006). “Computation of modal stress resultants for completely free vibrating plates by LSFD method,” Journal of Sound and Vibration, 297, 704-726. 4. Wang, C. M., Wu, W. X., Shu, C. and Utsunomiya, T. (2006). “LSFD method for accurate vibration modes and modal stress-resultants of freely vibrating plates that model VLFS,” Computers and Structures, 84, 2329-2339. 5. Shu, C., Wu, W. X. and Wang, C. M. (2006). “Least squares finite difference method for vibration analysis of plates,” Chapter of the book “Analysis and design of plated structures: Volume 2: Dynamics”, Edited by N. E. Shanmugam and C. M. Wang. Woodhead Publishing Limited, Cambridge England. 6. Shu, C., Wu, W. X., Ding, H. and Wang, C. M. (2007). “Free vibration analysis of plates using least-square-based finite difference method,” Computer Methods in Applied Mechanics and Engineering, 196, 1330-1343. 7. Wu, W. X., Shu, C. and Wang, C. M. (2007). “Vibration analysis of arbitrarily shaped membranes using local radial basis function-based differential quadrature method,” Journal of Sound and Vibration, 306, 252-270. 8. Wu, W. X., Shu, C., and Wang, C. M. “Application of meshfree least squares-based finite difference method to large-amplitude free vibration of arbitrarily shaped thin plates,” in press. 300 [...]... force Qy for 4th mode 263 Fig 8.10 Mode shape w for 5th mode corresponding to Ω = 3.3033 263 Fig 8.11 Bending moment M x for 5th mode 264 Fig 8.12 Bending moment M y for 5th mode 264 XXIII Fig 8.13 Twisting moment M xy for 5th mode 264 Fig 8.14 Shear force Qx for 5th mode 265 Fig 8.15 Shear force Qy for 5th mode 265 Fig 8.16 Mode shape w for 6th... equations gives all necessary information for calculating stresses at any point of the moderately thick plate 1.1.2 Analytical Methods for Plate Analysis The most ideal situation in solving the governing partial differential equation of plate is to find an exact analytical solution However, analytical solutions are only possible for very few simple types of loading, plate shapes and boundary conditions... plate vibrating in 4th mode 184 Fig 5.13 1st principal modal bending moments M x′ for circular plate vibrating in 4th mode 184 Fig 5.14 2nd principal modal bending moments M y′ for circular plate vibrating in 4th mode 185 Fig 5.15 Maximum modal twisting moments M x′′y′′ for circular plate vibrating in 4th mode 185 Fig 5.16 Maximum modal shear forces Qx′′′ for. .. modal twisting moments M x′′y′′ for elliptical plate ( a b = 2 ) vibrating in 4th mode 187 Fig 5.21 Maximum modal shear forces Qx′′′ for elliptical plate ( a b = 2 ) vibrating in 4th mode 187 Fig 5.22 Modal deflections W for lifting-tab shaped plate vibrating in 4th mode 187 XIX Fig 5.23 1st principal modal bending moments M x′ for lifting-tab shaped plate vibrating in 4th mode ... principal modal bending moments M y′ for lifting-tab shaped plate vibrating in 4th mode 188 Fig 5.25 Maximum modal twisting moments M x′′y′′ for lifting-tab shaped plate vibrating in 4th mode 188 Fig 5.26 Maximum modal shear forces Qx′′′ for lifting-tab shaped plate vibrating in 4th mode 189 Fig 5.27 Modal deflections W for 45° right triangular plate vibrating in 4th mode 189... bending moments M x′ for 45° right triangular plate vibrating in 4th mode 189 Fig 5.29 2nd principal modal bending moments M y′ for 45° right triangular plate vibrating in 4th mode 190 Fig 5.30 Maximum modal twisting moments M x′′y′′ for 45° right triangular plate vibrating in 4th mode 190 Fig 5.31 Maximum modal shear forces Qx′′′ for 45° right triangular plate vibrating in... supported square plate under the uniform load q = q0 = 10000N/m 2 216 Fig 6.11 Numerical bending moment M y (left) and numerical shear force Qx (right) of the simply supported square plate under the uniform load q = q0 = 10000N/m 2 216 Fig 6.12 Numerical shear force Qy (left) and numerical effective shear force Vx (right) of the simply supported square plate under the uniform load q... circular plate 257 Fig 8.3 Modal stress-resultants of 4th mode ( n = 2, s = 1 ) of circular plate 258 Fig 8.4 Mode shape w for 4th mode corresponding to Ω = 1.3557 261 Fig 8.5 Bending moment M x for 4th mode 262 Fig 8.6 Bending moment M y for 4th mode 262 Fig 8.7 Twisting moment M xy for 4th mode 262 Fig 8.8 Shear force Qx for 4th mode 263 Fig 8.9 Shear force... analyses of thin plates, large-amplitude free vibration of thin plates, free vibration of Mindlin plates, characteristic analysis of metallic waveguides, and free vibration of uniform membranes The complexities associating with these problems correspond actually to the difficulties (1) to (4) mentioned above in solving PDEs For example, the governing equations for plates in the classical thin plate theory... buildings They form bulkheads, decks, tanktops, bottoms, side panels, floors, deep girders, etc The flexural properties and behavior of a plate depend greatly on its thickness in comparison with its other dimensions Corresponding to this dependence, theories used in plate analysis may be classified into two types: thin plate theories and shear deformable (moderately thick) plate theories In plate analysis, . of Plate Analysis 1 1.1.1 Introduction to Plate Theories 1 1.1.2 Analytical Methods for Plate Analysis 4 1.1.3 Numerical Methods for Plate Analysis 6 1.2 Literature Review on Mesh-free Methods. Mesh-free Methods for Plate Analysis Wu Wenxin NATIONAL UNIVERSITY OF SINGAPORE 2007 Effective Mesh-free Methods for Plate Analysis . Differential Equation for Transversely Loaded Plates 109 4.1.5 Differential Equation for Freely Vibrating Plates 113 4.1.6 Differential Equation for Buckling of Plates 114 4.1.6.1 Plates under Combined